Cancel common factors in symbolic product

Suppose I have the following function

G[n_, k_] := Product[g[n + i]/g[i], {i, 0, k}] 

where $ n,k$ are assumed to be natural numbers and $ g$ is a nonzero symbolic function. I want to simplify the expression G[n,k+1]/G[n,k] and cancel all common terms to get g[n+k+1]/g[k+1]. However, Simplify, FullSimplify, Expand and Cancel all yield the symbolic (not cancelled) product.